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On zeros of Hecke \(L\)-functions and their linear combinations on the critical line. (English. Russian original) Zbl 1233.11097
Dokl. Math. 81, No. 2, 303-308 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 431, No. 6, 741-746 (2010).
Let \(f(z)= \sum^\infty_{n=1} a(n)e(nz)\) be a holomorphic cusp form of weight \(k\), level \(D\) and character \(\chi\). Moreover, denote by \(L_f(s)=\sum^\infty_{n=1}a(n)n^{-s+(1-k)/2}\) the corresponding (normalized) \(L\)-function. Then \(L_f\) belongs to the Selberg class of degree 2. In particular, it satisfies a Riemann type functional equation with one gamma factor relating \(F(s)\) to \(\overline{F(1-\overline s)}\). With this notation the following result is formulated.
Theorem 1: If \(f(z)\) is an eigenfunction of the Hecke algebra then a positive proportion of non-trivial zeros of \(L_f(s)\) lie on the critical line \(\sigma= 1/2\).
In the second part of the paper the author considers the problem of zeros of linear combinations of Hecke \(L\)-functions, focusing again on these with real parts \(1/2\). The set up is as follows. Let \(\psi_j\) be complex characters of imaginary quadratic field \(\mathbb Q(\sqrt{-D})\), and let \(a_j\) be real numbers. Moreover, let \({\mathfrak f}(s)=\sum_j a_j L(s,\psi_j)\).
Theorem 2: For a sufficiently large \(T\), the number of zeros \(\rho= (1/2)+ i\gamma\) of \({\mathfrak f}(s)\) with \(0<\gamma< T\) is \(\gg T(\log T)^{2/h} e^{-c\sqrt{\log\log T}}\), where \(h= h(-D)\) is the class number, and \(c\) is an absolute constant.
Only very rough sketches of proofs are provided. One can judge that the author is building on classical ideas by Selberg and Hardy with further developments by Karatsuba, Voronin and Jutila. Details are promised to be given in forthcoming papers [Izv. Math. 74, No. 6, 1277–1314 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 6, 183–222 (2010; Zbl 1233.11098) and Math. Notes 88, No. 3, 423–439 (2010); translation from Mat. Zametki 88, No. 3, 456–475 (2010; Zbl 1257.11083).

MSC:
11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
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